Global solution of the cauchy problem for a class of 2 × 2 nonstrictly hyperbolic conservation laws

We prove the existence of a global weak solution to the Cauchy problem for a class of 2 × 2 equations which model one-dimensional multiphase flow, and which represent a natural generalization of the scalar Buckley-Leverett equation. Loss of strict hyperbolicity (coinciding wave speeds with a (1101) normal form) occurs on a curve in state space, and waves in a neighborhood of this curve contribute unbounded variation to the approximate Glimm scheme solutions. The unbounded variation is handled by means of a singular transformation; in the transformed variables, the variation is bounded. Glimm's argument must be modified to handle the unbounded variation that appears in the statement of the weak conditions, and this requires that the random choice variable be random in space as well as time